The Foundations of Science · Henri Poincaré
VIII
_The Demonstrations of the Principle of Induction_ Let us now examine the pretended demonstrations of the principle of induction and in particular those of Whitehead and of Burali-Forti. We shall speak of Whitehead's first, and take advantage of certain new terms happily introduced by Russell in his recent memoir. Call _recurrent class_ every class containing zero, and containing _n_ + 1 if it contains _n_. Call _inductive number_ every number which is a part of _all_ the recurrent classes. Upon what condition will this latter definition, which plays an essential rôle in Whitehead's proof, be 'predicative' and consequently acceptable? In accordance with what has been said, it is necessary to understand by _all_ the recurrent classes, all those in whose definition the notion of inductive number does not enter. Else we fall again upon the vicious circle which has engendered the antinomies. Now _Whitehead has not taken this precaution_. Whitehead's reasoning is therefore fallacious; it is the same which led to the antinomies. It was illegitimate when it gave false results; it remains illegitimate when by chance it leads to a true result. A definition containing a vicious circle defines nothing. It is of no use to say, we are sure, whatever meaning we may give to our definition, zero at least belongs to the class of inductive numbers; it is not a question of knowing whether this class is void, but whether it can be rigorously deliminated. A 'non-predicative' class is not an empty class, it is a class whose boundary is undetermined. Needless to add that this particular objection leaves in force the general objections applicable to all the demonstrations.
Burali-Forti has given another demonstration.[16] But he is obliged to assume two postulates: First, there always exists at least one infinite class. The second is thus expressed: u[epsilon]K(K - [iota][Lambda]) · [inverted c] · u < v'u. The first postulate is not more evident than the principle to be proved. The second not only is not evident, but it is false, as Whitehead has shown; as moreover any recruit would see at the first glance, if the axiom had been stated in intelligible language, since it means that the number of combinations which can be formed with several objects is less than the number of these objects. [16] In his article 'Le classi finite,' _Atti di Torino_, Vol. XXXII.
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